and where \(W(Y,\hat{Y}):=Y^T\!J\hspace*{1.0pt}\hat{Y}\)  is the Wronskian of \(Y=(X^T,U^T)^T\)  and
\(\hat{Y}=({\hat{X}}^T,{\hat{U}}^T)^T\) , following the notation in (REF ). The dagger
in (REF ) denotes the Moore–Penrose pseudoinverse, see e.g.
[1]}, [2]}. The notation \(\operatorname{\text{\rm ind}}{\mathcal {P}}\)  means the number of negative
eigenvalues of the symmetric matrix \({\mathcal {P}}\) . Note that \(\operatorname{\text{\rm ind}}{\mathcal {P}}+\operatorname{\text{\rm ind}}\,(-{\mathcal {P}})=\operatorname{\text{\rm rank}}{\mathcal {P}}\) .
For convenience we also define the constant \(2n\times n\)  matrix \(E\)  representing
the vertical Lagrangian plane, i.e.,
\( E:=\left(\begin{matrix} 0 & I\end{matrix}\right)^T.\) 
